Let (M,ℱ) be a foliated manifold. We describe all natural operators lifting ℱ-adapted (i.e. projectable in adapted coordinates) classical linear connections ∇ on (M,ℱ) into classical linear connections (∇) on the rth order adapted frame bundle $P^r(M,ℱ)$.

A classification of all $ℳf_m$-natural operators $A:G{r}_p⟿G{r}_{q}T*$ lifting p-dimensional distributions D ⊂ TM on m-manifolds M to q-dimensional distributions A(D) ⊂ TT*M on the cotangent bundle T*M is given.

We classify all natural operators lifting linear vector fields on vector bundles to vector fields on vertical fiber product preserving gauge bundles over vector bundles. We explain this result for some known examples of such bundles.

We classify all natural operators lifting projectable vector fields from fibered manifolds to vector fields on vertical fiber product preserving vector bundles. We explain this result for some more known such bundles.

We describe all ${}_m\left(G\right)$-gauge-natural operators lifting right-invariant vector fields X on principal G-bundles P → M with m-dimensional bases into vector fields (X) on the rth order principal prolongation $W^{r}P=P^{r}M{\times }_M{J}^{r}P$ of P → M. In other words, we classify all ${}_m\left(G\right)$-natural transformations $J^{r}LP{\times }_M{W}^{r}P\to T{W}^{r}P=L{W}^{r}P{\times }_M{W}^{r}P$ covering the identity of $W^{r}P$, where $J^{r}LP$ is the r-jet prolongation of the Lie algebroid LP=TP/G of P, i.e. we find all ${}_m\left(G\right)$-natural transformations which are similar to the Kumpera-Spencer isomorphism $J^{r}LP=L{W}^{r}P$. We formulate axioms which characterize...

Let m and r be natural numbers and let $P^r:ℳf_m\to ℱℳ$ be the rth order frame bundle functor. Let $F:ℳf_m\to ℱℳ$ and $G:ℳf_k\to ℱℳ$ be natural bundles, where $k=dim\left(P^r{ℝ}^m\right)$. We describe all $ℳf_m$-natural operators A transforming sections σ of $FM\to M$ and classical linear connections ∇ on M into sections A(σ,∇) of $G\left(P^{r}M\right)\to P^{r}M$. We apply this general classification result to many important natural bundles F and G and obtain many particular classifications.

We describe all natural operators lifting nowhere vanishing vector fields X on m-dimensional manifolds M to vector fields (X) on the rth order frame bundle $L^{r}M=invJ{₀}^r({ℝ}^m,M)$ over M. Next, we describe all natural operators lifting vector fields X on m-manifolds M to vector fields on $L^{r}M$. In both cases we deduce that the spaces of all operators in question form free $(m(C_r^{m+r}-1)+1)$-dimensional modules over algebras of all smooth maps $J{₀}^{r-1}T̃{ℝ}^m\to ℝ$ and $J{₀}^{r-1}T{ℝ}^m\to ℝ$ respectively, where $C{ⁿ}_k=n!/(n-k)!k!$. We explicitly construct bases of these modules. In particular, we...

Let $r,n$ be fixed natural numbers. We prove that for $n$-manifolds the set of all linear natural operators $T^{*}\to T^{*}{T}^{\left(r\right)}$ is a finitely dimensional vector space over $R$. We construct explicitly the bases of the vector spaces. As a corollary we find all linear natural operators $T^{*}\to T^{r*}$.

Let $J^{r}T*M$ be the r-jet prolongation of the cotangent bundle of an n-dimensional manifold M and let $(J^{r}T*M)*$ be the dual vector bundle. For natural numbers r and n, a complete classification of all linear natural operators lifting 1-forms from M to 1-forms on $(J^{r}T*M)*$ is given.

For natural numbers $r$ and $n\ge 2$ all natural operators $T_{|ℳf_n}⇝T^{*}\left(J^r{T}^{*}\right)$ transforming vector fields from $n$-manifolds $M$ into $1$-forms on $J^r{T}^{*}M=\{j_x^r\left(\omega \right)\mid \omega \in {\Omega }^1\left(M\right),x\in M\}$ are classified. A similar problem with fibered manifolds instead of manifolds is discussed.

The second order transverse bundle $T_{}^{2}M$ of a foliated manifold $M$ carries a natural structure of a smooth manifold over the algebra ${𝔻}^2$ of truncated polynomials of degree two in one variable. Prolongations of foliated mappings to second order transverse bundles are a partial case of more general ${𝔻}^2$-smooth foliated mappings between second order transverse bundles. We establish necessary and sufficient conditions under which a ${𝔻}^2$-smooth foliated diffeomorphism between two second order transverse bundles maps...

We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on $T^{A}M$, where $T^A$ is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.